# Scientific journal European Journal of Natural History ISSN 2073-4972

### A VERY INEXPENSIVE SCHEME ON RFEM TO USE IN CFD AND OTHER PROBLEMS

1 The Office of Counseling and Research Fluid Engineering and Aerodynamic
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Note, Reference [1] should be read before reading this paper. In this paper I am going to use the Reduced Finite Element Method (RFEM), see [1] with a constant amount of Element Degree Limiter, Field Variable Limiter (EDL, FVL) coefficient to obtain an inexpensive scheme of non-oscillatory, for this purpose I used third degree Lagrangian elements on full upwind difference scheme (FUDS) and its result was very successful.
Reduced Finite Element Method
Non-oscillatory
Full upwind difference scheme
Third-order scheme
1. Khaleghi M.R.A. A new computational package for using in CFD and other problems.

The non-oscillatory shape functions [1] is written as follows:

(1)

Where is Element Degree Limiter, Field Variable Limiter (EDL, FVL), is p-order shape functions (in this paper degree of p is 3) and , and are reference functions I used equation (40) of [1] as reference function that is

(2)

Fig. 2. 1D convection-diffusion equation

Fig. 3. Grid for flow over a NACA 0012 airfoil

Fig. 4. Pressure distribution on surface of NACA 0012 airfoil, steady-state

Fig. 5. Grid for flow around a cylinder.

Fig. 6. Flow around a cylinder, steady-state

Where and are liner shape functions, by putting equation (2) in (1) and also we have

(3)

Equation (3) is non-oscillatory shape functions that will be used in this paper for CFD problems.

Examples

In this section, I am going to use the equation (3) as shape function for approximating a few equation and compare its result with non-constant . As first example, I approximated 1D advective equation by Point Collocation weight function, see figure (1). In second example, I approximated 1D convection-diffusion equation in by Galerkin weight function, see figure (2). In third example, I approximated 2D Euler equations for pressure distribution on surface of NACA 0012 airfoil [1] by Galerkin weight function and infinite elements for non-solid boundaries, see figure (4), and flow around a cylinder [1] by Subdomain Collocation weight function see figure (6).

Conclusions

As can be seen from the results, with any number of elements solutions are non-oscillatory, and when we use fine mesh (usually in CFD is used) solutions are very close together. So, using of constant EDL, FVL is useful.